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L
THE THEORY
OF THE
CONSTRUCTION OF ORDNANCE
WITH SPECIAL REFERENCE TO THE
RESISTANCE OF GUNS
TO
TANGENTIAL STRAIN.
PEEPAEED FOE USE AT THE U. S. NATAL ACADEMY.
ANNAPOLIS, MD,
1883.
TTASHINaTON:
GOYERNMENT PEINTINa OFFICE.
1883.
5318 1
INTRODUCTION.
The following pages consist principally of a translation of extracts
from Yirgile's work on the "Eesistance des tubes metalliques, simples
ou composes, avec application a la construction des benches a feu."
The deduction of the equations in Part I, which are used in the ap-
plication of the theory in Part II, has greatly simplified the discussion
of resistance to tangential strain. This part of the work has been done
by Lieut. C. A. Stone, and is used instead of Yirgile's method.
Some few changes and omissions have been made in the original, as
the object in view was simply to provide a suitable text for purposes of
instruction.
H. B. EOBESOK,
Commander JJ. S, H.
3
THE THEORY OF THE CONSTRUCTION OF ORDNANCE.
FIRST PART.
THEORY.
CHAPTER I.— GEIS^EEAL PEIKCIPLES.
1. Let a right rectangular prism be submitted to a tension parallel to
one of its systems of edges so tliat the strain shall be uniformly spread
over the transverse section in the proportion of iJ units of force for a
unit of surface. The edges on which the force acts will be lengthened,
while those that are perpendicular to the direction of the force will be
shortened.
While the force p does not exceed a certain limit, the changes of form
which it determines will not be of a permanent character ) they will
disappear when the force ceases to act. It is then said that these
changes of form are elastic.
If the force j? exceeds this limit, the prism, after the force has ceased
to act, will not exactly resume its former dimensions. The changes of
form thus produced are said to be permanent. Permanent changes of
form, not apparent at first, increase more rapidly than the changes due
to elasticity. A moment even occurs when the latter diminish and dis-
appear almost entirely at the instant of rupture.
If the force _p is a pressure instead of a tension, the contrary effects
will be observed, that is to say, the edges parallel to^ will be shortened
while the others will be elongated. Otherwise the prism will pass
through the same changes of form, at first elastic, then i^ermanently
changed and elastic at the same time, at last almost entirely i^erma-
nently changed, and this series of phenomena will be terminated by the
crushing of the prism.
2. The value of j? for which these changes of form cease to be entirely
elastic is called the limit of elasticity. There is for each substance and
for each particular state of this substance two limits of elasticity, one
referring to tension and the other to pressure.
When ]) does not exceed the limit of elasticity, the elongation or
compression determined by this force is proportional to it, whether
applied in its own direction or at right angles 5 but beyond this limit
the change of form increases more rapidly than p. In theory it is not
considered that the pressures or tensions exceed the limit of elasticity.
This reservation admitted, let i be the elongation or compression pro-
duced by the force p acting in the direction of the elongation or com-
pression 5 1^ is called the modulus of elasticity; this ratio is ordinarily
1
expressed by E.
The modulus of elasticity is the force, theoretically, which must be
applied to a unit of transverse section to double the length of the prism.
3. Wertheim, by experimenting with diaphanous bodies, found that
the modulus of elasticity of extension was the same as for compression.
We take for granted that it really is so. At the same time it is to be
remarked that the experiments of physicists give smaller values of the
modulus of elasticity, in the case of certain metals, for compression than
for extension. These differences may be caused by the manner of action
of the forces which govern the constitution of the bodies experimented
upon.
4. Wertheim found by a series of carefully conducted experiments
that an actual limit of elasticity did not exist ; that is to say, that all
strains, no matter how feeble, provided that they are of long duration,
cause a permanent elongation. On the other hand many authors believe,
and it is generally conceded, that the changes of form increase more
rapidly than the forces which cause them, even when they are far from
the limit where a permanent change of form is easily distinguished; so
that there is, strictly speaking, no actual modulus of elasticity.
However, it is certain, as we have already seen, that permanent
changes of form are at first extremely small in comparison with the
changes that are elastic; at least this is true for all bodies which may
be termed in any degree elastic. It is equally true that the curve of
the change of form as a function of the force which produces the change
is, to a greater or less extent, identical with the tangent at the origin
of co-ordinates. It therefore results that we may consider the theory
based upon the existence of a limit and modulus of elasticity, as exact
for small tensions and pressures, but as susceptible of furnishing only
approximate results for greater values. The strain or pressure to which
this theory can be applied without too great an error will be the prac-
tical limit of elasticity , and the mean modulus corresponding to the press-
ures and tensions comprised in this limit will be taken as the modulus
of elasticity.
Let E = the interior radius of the tube ;
E'= the exterior radius of the tube ;
P = the distance of any point from the axis ;
t = the hoop-tension at any point ;
p = the radial pressure at any point ;
T and T', P and P^ the values of t and pforp = B> and p = IV. Then
P and P' will be the internal and external pressures.
7ksm*0
9 ^^ »
-- .T^'^-TK.
Consider the section aa'^ through the axis of length
I. The section of the metal on either side may be gene-
rated by the line I moving parallel to the axis, with
the rate dp from />=R to p:=W. Therefore the re-
sistance of the tube to rupture along the given section
will be
2i£ tap
The bursting effort will be 2 Z (PE — Y'W) ; therefore
/ ^^^ = PE~P^R' .
(1)
Let t=f'(p)) substituting this value of t in equation (1), and
integrating, we have
/ (EO — / (E) = P E — P^ E'.
Since E and E' are independent of each other, we can write
f(W) = -V'n'
f{p) = -PP'
Differentiating this equation, with reference to /?, gives
or in general
/'(,)=-j,-,|=*
(2)
The solution is effected by means of two partial solutions. First:
d 1)
Let— ^=m=a constant in (2), then 7^=0, and equation (2) becomes
t——p=m. Substituting this value in (1) we have
m / ''^/>=PE-P^E'=m(E'-E),
which satisfies the equation.
For the second solution, let t=p in equation (2). Then that equation
becomes
P dp
Let
then
2 dp
dp_^2a
'dTp^y
a
(3)
(4)
8
Substituting this value of t and^? in equation (1),
•/■
dp _ r 1 1 1
Integrating, we have
1 r 1 111 1
c-2 LR^-^ E''^-^ J~ E^-2'" E'--2*
To satisfy this equation we make c=3; which gives in equation (4)
a
The value of c may also be determined by substituting the value of
a from (4), a =jt?/>'^^ in equation (3), and integrating j
therefore
log|=2 1og^,
whence
Therefore p varies inversely as the square of p. Adding the two
partial values of *, we have
and also for^,
t = ~-^m (5)
P = -,-m (6)
From equations (5) and (6) we have,
f-jp = 2m, * + ^ = 7? ..... (7)
Therefore
(^ + i>)^' = (T+P)R' (8)
and ^_^=:T-P (9)
From equations (7) the following properties are deduced :
the difference between the hoop-tension and the radial pressure is
the same at each point ) and the sum at any point of the hoop-tension
and the radial pressure varies inversely as the square of the distance
of the point from the axis.
^'CU, [aXTiaxV.
To determine a, substitute the value of c in equation (3), and integrate,
and we have
Let i? = P^ and ^ = E^ ; then
a =
(P - PQ E^^ B^
W -E^
(10)
Substituting this value of a in the preceeding equation, solving for
^, and reducing, we have
PE^(E^^-^^) + P^E^^(^^-E^) ...
To determine m : From (6)
substituting the value of a from (10), and reducing,
PE2-P^E'2
. W^-H^
(12)
PAET II.— CHAPTER I.
APPLICATION OF THE THEORY.
Tubes and guns consisting of one piece of metal.
§ 1 . The equations deduced in Part I, although they suppose the
elasticity of a body to remain constant in all directions, will answer for
most cases that occur in practice. It is, however, true that the elas-
ticity of the same body varies both as regards direction and position,
though these variations are generally inconsiderable when applied to
metals.
2. — Resistance of homogeneous tubes.
Assuming equations (8) and (9) of Part I,
(^M t-p =T-P
in which T and P represent the tension and the pressure at the surface
of the bore, t and p the tension and pressure of any point situated at
a distance p from the axis, E the radius of the bore. If T' and P' repre-
sent the tension and the pressure at the exterior surface, and E^ the
radius of this surface, we shall have, putting p = E^,
w{
(T' + P0E^2^(T + P)E2
T'_P/ =:T-P.
Combining (a) and (b), we derive:
i{t+p)p^={T + ^^)n-
^ M t-p = T^-P^
Since P' only stands for the pressure of one atmosphere, it may be
neglected. Putting, then, P^ = 0, eliminating T, T^, snidp in the four
equations {a) and (c), and solving with reference to t, we have:
E'2
^ ~ ^ E^2
We see by this formula that the maximum value of t corresponds to the
minimum value of p', that is to say, of ^ = E. It is then at the surface
of the bore that the metal undergoes the greatest tangential strain. It
results from this that we will assure the resistance of the tube (to
10
11
racture) by determining the pressure P so that the tension T, at the
surface of the bore, shall not exceed the tensile strength of the metal.
3.—Calculatio}i of resistance.
Putting ^ = R, the preceding formula beconfes
R'2 ^
-R^ + 1
T = P^^,--
R^ ~ ^
from which we have
(1) P=T ^^ ^
R^24.R2
This result may be obtained directly by making P' = 0, in equations
(&), by eliminating T' and solving with reference to P.
Formula (1) was given for the first time by Lame ; it readily permits
the calculation of the pressure P, which corresponds to a given tension
of metal. If T is the limit of tension which is not to be exceeded, P will
represent the resistance of the tube, these two quantities being referred
to the unit of surface.
If in formula (1) we make E' vary from the minimum value of R togl^
or else R from R' to 0, P will vary from to T. Thus if T represents the
greateat tension which the metal is capable of sustaining without im-
pairing its elastic qualities, the pressure P of the gas in the interior of
the bore should never ^ceed T. This principle was established by
Lame, and afterwards followed by all other authors who have consid-
ered the resistance of hollow cylinders.
4. — Application to ordnance.
The thickness of old cannon at the breech and re-inforce, that is, at
the part exposed to the greatest tension of the powder gas, is usually
about one caliber or twice the radius of the bore. If we introduce this
quantity in formula (1), R^^ 3 R and P = 0.8 T. Thus in increasing the
thickness of the metal from one caliber to infinity, the resistance of the
gun is only increased from 0.8 T to T, that is to say, one-quarter of its
original value.
If a great weight of metal is to be disposed of, it is better to lengthen
the piece, or better still to increase the caliber. In either case we at-
tain better results from the powder and may hope to develop sufficient
energy in the projectile to accomplish our purposes without unnecessarily
straining the gun.
Let us take, for example, an old smooth-bore gun of 30, of which the
caliber is 164°^™.7, and the thickness at the re-inforce about 1.17 of the
caliber, or 2.34 of the radius of the bore; it will require to be made of
excellent metal to sustain a prolonged fire witli a projectile of 15 kilo-
12
*
grams, and a charge of i)owder of 5 kilograms, equal to one-tliird the
weight of the projectile. By reaming out the bore of this gun 8"^°^.2,
that is to say -^ of the radius of the bore, the gun is converted into a
piece of 40. If by the same process 21™™ of metal be removed, we shall
have a piece of 60. The pressures which these three guns will sustain
will be, respectively,
0.82 T, 0.79 T, and 0.73 T.
The resistance of the gun of 60 will be but ^ less than that of the
smaller piece of 30.
Let it be supposed that the charge remains fixed at 5 kilograms of
I)Owder for the three guns; the guns of 40 and 60 will fire, respectively,
spherical projectiles of 20 and 30 kilograms with normal charges of J
and ^; the gun of 40 will present a much greater security than that of
30, and all danger of explosion will have disappeared in the gun of 60.*
THE RODMAN PROCESS.
Let a gun having a thickness at the re-inforce of one caliber be taken
for discussion. If, by any means whatever, each part of the metal in
the whole thickness of the gun can be made to do an equal share of
rk, the pressure that may be developed in the bore will be represented
the formula
r
from which we deduce by making R = 3r, P=:2T; that is to say, that
the resistance will be double that of a cylinder of unlimited thickness
and cast in the ordinary manner.
Major Eodman has realized this principle, or at least partially so, by
a peculiar process of fabrication which has been applied to guns of large
caliber.
These guns are cast on a hollow core, and the mold is placed in a fur-
nace. After the casting, a current of water circulates in the core while
the flask is heated on the exterior. The result of this arrangement is,
that the central parts of the casting are first cooled and solidified. After
the whole mass becomes solid, the exterior layers are more expanded
than those near the bore, and after the cooling they remain distended,
and exert a compression on the inner layers, which aid the latter in re-
sisting the pressure of the powder-gas.
Experiment shows that the Eodman process effectively increases the
resistance, but it is evident that its efficiency can only be partial. In
order that the result should be complete, it would be necessary that the
*Tlie autlior in a note fnrtlier states that it is not his intention to especially rec-
ommend the construction of guns of large caliber. Other considerations enter into
the construction of modern artillery. The article, however, is none the less impor-
tant as it illustrates a useful principle. — H. B. R.
13
gun should cool aud solidify regularly from the interior to the exterior,
and this cannot take place. The metal which forms the flask of the
mold, although very much heated, must remain at a temperature lower
than that of the casting 5 otherwise it would melt, or, if it were refrac-
tory, dilate and detach itself from the sand, causing the mold to break
up under the pressure of the liquid metal. It follows, therefore, that
the layers of metal in contact with the mold will be solidified, while
those nearer the bore will still remain in a liquid state.
Q.— Effect of vibrations.
If the pressure of the powder-gas is slowly developed, so that at each
ifistant the equilibrium between this pressure and the elastic resistance
of the different layers of metal is maintained, the work of the gas at
each instant will be destroyed by the work of resistance due to elastic-
ity, and the two forces will be in equilibrium. If, on the contrary, the
pressure of the gas is suddenly developed at its maximum, and is main-
tained for an appreciable time, the work of the gas will be in the first
instants greater than the resistance of elasticity, and vibration will be
produced.
If we let i be the extension or expansion of the surface of the bore
in the case of static equilibrium, and e half the amplitude of the vibra-
tory movements, i and e being referred to the unit of length, ^ 4- e will
be the greatest expansion of the bore. In the case of static equilibrium
this expansion, i-\-e^ will be determined by a tension equal to E (i -j- e) ;
but, according to the hypothesis, the same expansion is caused by the
tension Ei We have, then, a right to conclude that the resistance of
a gun is diminished in the ratio of i + e to i, since it is from the expan-
sion of the bore that the strain on the gun may be determined.
If the transmission of force across the walls of a tube were instanta-
neous, the resistance of elasticity at each instant would be expressed
by the formula (1), in which P and T are variables. Calling y this resist-
ance and X the corresponding expansion of the bore^ referred to the
unit of length, we shall have
The distance passed over by, this resistance in the passage of the ex-
pansion X to x-^-dx being EcZa?, the work developed in this small move-
ment will be, for each unit of surface of the bore,
and the total work for the expansion will be
I I ydx=
R I ydx= ^n,^2 I^E Y^
14
During the same displacement of the surface of the bore, the force
P produces a motive force equal to PE x; as this force is expressed
we shall have
PE X= T>/2 I T?2 ^^ ^^»
The living force developed at the outset of the movement will be en-
tirely extinguished when the motive force equals the total resisting
force just found j whence it is easily seen that
x=2i. '
In reality, the two hypotheses just given j the instant transmission
of force, and the complete destruction of the living force, are not true,
and the tubular envelope, strictly speaking, is never in equilibrium.
The exterior surface is always a little behind the interior surface. It
is consequently less distended than we have supposed in the formula
(1), and the evaluation of the resisting force is a little too great. On
the other hand, the same resisting force is a little less than the motive
force, for when the movement of expansion is finished on the interior
or surface of the bore it still continues on the exterior as a small portion
of the living force. Thus in supposing equality between the two ex-
pressions
Wm:e^^^^^^^e-+e^^2
two errors are committed in an inverse sense. But these errors are
very small, and taking into consideration the rapidity with which
motion is transmitted from the interior to the exterior of the tube, we
may write ^ = 2i as sufficiently exact. This equation signifies that :
The expansion of the tore of a gun produced hy a pressure suddenly de-
veloped is double that produced by a progressive increase of the same pressure.
In practice the pressure of the powder- gas is not developed instanta-
neously, though rapidly, so that the value of i + e will be found be-
tween i and 2i.
7. — Influence of grooves.
In a rifled gun the pressure of the gas is not upon the surface of the
bore, but upon a cylinder tangent to the bottom of the grooves. It has
been shown how little the resistance is diminished by increasing the
bore by a small quantity j now,'^in a gun of a caliber of 16 centimes the
depth of the grooves does not exceed the 0.073 of the radius of the bore,
and for other guns it is still less. In such a case but a small error will
be committed in neglecting the depth of the grooves. But there exists
in guns deeply grooved another cause of destruction. This is the effect
15
of the powder gas wMch enlarges the grooves by compressing the
lands which separate them, causing the sides of the groove to separate
from the bottom, and giving rise to cracks and fissures in the re-enter-
ing angle formed by the junction of the surfaces. This cause of de-
terioration is rapid and has been very marked in many systems of rifled
ordnance. Hence the necessity of diminishing as much as possible the
depth of the grooves and joining the principal lines of their profile by
means of curves.
8. — Effect of guns becoming heated.
Many persons regard the elevation of temperature resulting from a
rapid and prolonged fire as one of the causes of the bursting of guns.
^N^othing authorizes this presumption j on the contrary, it may be said
that the layers of metal which are near the bore become exijanded by
the heat more than those that are situated near the exterior, distending
the latter, which compress the inner layers and render the danger of
rupture less imminent.
The lowering of the modulus of elasticity also acts in the same way.
9. — Observations relative to permanent strains.
When the interior layers of a metallic tube are submitted to a tension
greater than their limit of elasticity, the strained parts which remain
elastic are alone proportional to this tension, the strains are permanent
and increase, consequently the exterior layers are more distended than
they would have been without the intervention of the permanent strains,
and thus support the interior layers. It results from this, that formula
(1) gives in this case, too small a value to P, and that the difference be-
tween these values and the actual resistance of the tube increases as the
permanent strains are greater.
Thus two metals may equally resist a given tension ; one remains
perfectly elastic, the other undergoes a permanent strain j and if they
are used in the fabrication of two equal tubes, the tube made of the
first metal will have a resistance inferior to that of the other, and the
superiority of the former will be more marked as the permanent strain
is the greater. These considerations explain the great resistance of old
bronze cannon as compared with those made of cast iron.
Resistance of old smooth-hore guns.
It was found by comparing the initial velocities of projectiles and the
the velocities of recoil furnished by guns of different lengths of bore,
that an empirical formula might be established, by the aid of which
the pressure of the gas in the bore might be computed within a certain
degree of approximation. This formula applied to a gun of 30, throw-
ing a projectile of 15 kilograms, with a charge of J, gives 830 kilograms
for each square centimeter as the pressure of the gas.
If we introduce this quantity in formula (2) that we shall find further
on, and make W= 3.17 E, we find T = 10 kilograms per square milli-
meter. '
A
CHAPTEE II.— BUILT-FP TUBES AND GUNS.
1. Co7iditions of resistance of an elementary tube.
A built-up tube is composed of several simple tubes placed concen-
trically one over the other. These tubes are called the elementary tubes.
A built-up gun may be regarded a built-up tube. In order to make
a built-up tube, the elementary tubes are put together successively from
the center to the circumference; each one of them is reamed out to a little
smaller diameter than is necessary for its adjustment. The tube is
placed after its expansion by heat so that each one in cooling exercises
a strong compression on the inner one next it. It results from this ar-
rangement that the elementary tubes are tangentially compressed next
the bore of the gun, while those on the exterior are distended.
When the entire system is submitted to an interior pressure the
compression of the elementary tubes diminishes, disappears even, and
sometimes changes to expansion if the interior pressure is sufficiently
raised, while the exterior tubes, already distended, are still more strained.
The exterior tube has one of its two cylindrical surfaces free, and
follows the law of homogeneous tubes subjected only to an interior
pressure, while all the other tubes are subjected to two pressures, the
one interior and the other exterior. The formulas which relate to the
latter are deduced from the two equations (b) of Chapter I, Part II, as
follows :
(T+P)R2=(T'4.p/)R/2, T-P=T'-P'-
eliminating T', we have
m T-p^Ljt^'-p/^^E^
^^ W^-n^ B'^-W
Let one of the elementary tubes submitted to two pressures be con-
sidered.
If t be called the tension of the metal in the direction of the circum-
ference of a point situated at a distance p from the axis, and p the press-
ure acting on the same point in the direction of the radius, we shall
have Chapter I, Part II, equations {a) and (c)
(T+P)R2=(^-fj)>2 T-P=*-i>
{t+p)p^={T'+-p')W' t-'p=T-V\
16
17
Eliminatiiig T, T', iiiid j; from these four equations, we have
R^ n-+p^ R- E^+/ ^- " f; -^ '^
2+1 ",+1
'R^ R^2
IP
Differentiating, we have
If P is smaller than P', the formula (2) shows that T is negative,
that is to say, that the interior snrface is compressed; ^^ is then positive;
consequently the algebraic value of t increases with ,o, which sionities
that the compression goes on diminishing from the interior to the
exterior of the tube.
If we make />=R', then t has a negative value, which signifies that
the whole thickness of the tube is in a state of compression.
If P is equal to P^, T is still negative, but ^ reduces to zero, what-
ever value may be given to ^ -, in other words, all the circular fibers are
equally compressed.
2 P' R/2
If P is greater than P', but smaller than p/2T:p2? T is still negative,
as is T ; there is still compression of the interior surfa<^e; but this
compression increases from the interior to the exterior of the tube,
where it attains its maximum.
2 R'^
Finally, if P is greater than P^ ^ 09, T is positive, the interior
K^ -{- R~
surface is expanded; but- is negative, and it follows that the ten-
dp
sion of the fibers diminishes as the distance from the interior surface
increases.
It results, therefore, from this discussion, that the maximum tension
of the fibers (if there is any tension) is always on the interior surface;
consequently it is necessary, in order that the elementary tube shall
withstand the interior pressure, that T shall never exceed the tensile
strength of the metal of which the tube is formed. For the sake of
brevity, in speaking of the tension and limit of elasticity of a tube, the
tension and limit of elasticity of its interior surface will be undeistood.
5318 2
\ ^
18
Calculation of the resistance of a hiiilt-iq) tube.
Let Eo be the radius of the bore of a built-up tube, Ej, E,, E3 . . . E,„_i
the radii of the surtaces of the contact of the several elementary tubes
and E^ the exterior radius ; Po, Pi, P2 . . . P^, the pressures which act
normally to the cylindrical surfaces whose radii are Eo, E^, . . . E„ ;
Po is the pressure which the tiibe resists, and is the quantity to be de-
termined. In the general case P„, is the atmospheric pressure and will
be neglected.
Let us also call To, Ti, T2 . . . T^_i, the tensions of the circular fibers
composing the interior surfaces ofthe elementary tubes, and To', T/jTz' . . .
T'^_i the tensions of the exterior surfaces of the same tubes. A built-
up tube will support the greatest i)ressure when all the elementary
tubes are strained at the same time to tlieir limits of elasticity respect-
ively. Suppose this condition to be fulfilled, Tq, Ti, T2 . . . T^_i will
represent this limit for the tubes respectively corresponding to these
tensions. This being granted, and considering the two general equa-
tions
(T-fP)E2=(T'+p/)E'2, T-P=T'-P'5
we deduce, eliminating T' and solving with reference to P,
This formula applied to a tube occuj)ying a place in order of 11 + 1,
becomes
By making n=m and P^=0, we obtain P^_i. ]S"ow, making n=r/i— 1,
we shall have P;«_2. Knowing P^_2, T^_3, E^_3 and E«_2, P^_3 can be de-
termined in the same manner. In continuing thus, we shall finally ar-
rive at Po, which will be the resistance of the whole system, represented
by the greatest pressure that it will bear with safety.
It may be remarked that:
rt •p2 T>2 "f>2
In making this substitution in formula (3), it becomes after reduction :
This formula is more convenient for practical work.
Eeferring to formula (3), if ve make 7i=0 and Pi=0, it becomes:
E^o
E^ + E^o
Po— To-p2 , T>2 }
19
P<5 represents the resistance of the central tnbe alone and deprived of
the supiiort of the exterior tnbes.
If we make )i = and not Pi, we have:
biit we have, niakino- n
3 _T ^"
- J-O p2
)
=1,
P.=T,g
I
2E22
Substituting this valne in the expression for Po, we have
Jr-O-lo JJ2^_^J^2^+ J-l JJ2^_^iJ2^ R2^_^1^2^ •+--^2^^4 E^, E^^ + EV
an expression in which the first term represents the resistance of the
central tnbe, the second the fraction of the total resistance furnifched by
the second tnbe, and the thii^d the fraction of this resistanc^^ to all
the tubes exterior to the second. II
Eeplacing P2 by its valne as a function of T2, of P3, of E2 and of E3,
and continuing in the same way, we shall have a development of Pq in
which each term will represent the part of the resistance furnished by
one of the tubes of a built-up gun:
. p _^ E-\-E\, E^^-E^ _^E^_
^'^^ • . • • ^0--L0j^2^_^J^2_+J-l_^2^_^Ji2^ E\ + E'o
^rr. Fv^3-E\ 2E^ 2E^2
+ ^^E^34-E-2 E-;+K% E22+E^i
^^ E ^4-E^. 2R^ 2E^2 2E^3
-^-^^E^4+E^'3 E^o+E-1 E^2+E^ E23+E\
o. E2,.^-E2„ 2E^i m\_ ^E^+i^
"E^.+i+E^ E^o+E^ E^+E^s • • • E^.+i+E2"n
^^ 2E^ 2W, 2E^+i
E^o+E^ E^-hE^'2 E^+E^„+l
§ S.—EXA^riPLES OF APPtlCATIOW v,
1. — Cast-iron guns Iwojyed icitJt steel. (
Consideration of a cast-iron gun having the reinforce strengthened
•with hoops of puddled steel. From experiments made at the foundries
of ]S"ever.s and the forges at Guerigny, it was found that bar puddled steel
of the sa!ne quality as used for guns had a tensile strength of about
forty kilo-rams per square millimeter without contracting a permanent
elongation, and broke with a load of about 55 kilograms 5 but made in
20
the form of a tube for a gun, the same metal had only an elastic resist-
ance of from 2;") to 30 kilograms. The lower figures will be adopted^
say 25 kilograms.
As for the cast iron we will suppose its resistance reduced to 8 kilo-
grams per square millimeter, in this particular case, in order to take
into consideration the influence of the grooves and other causes of
deterioration, which may result from its immediate contact with the
powder-gas and with the projectile. This being understood, let it be
granted that the total thickness of the gun is one and a half calibers, a
caliber of cast iron and half a caliber of steel.
We have from these hypotheses :
Ej = 3Ko; andE2 = 4Eo.
The pressure, l*i, exerted b^^ the hoops on the cast iron, at the moment
when the powder-gas develops the greatest pressure, is giA^en in for-
mula (1), which in the case before us becomes:
_^z 32
Pi=Y2^-r--o2x25=7 kilograms per square millimeter,
and the interior pressure by formula (4) becomes
32 I
P(i=o2_r'] X (^^+^) + ^=19 kilograms per square millimeter,
or, after the mode of expression in general use, 1900 kilograms per
square centimeter. This represents the resistance of the gun.
A gun made entirely of cast iron, cast in a single piece, would have
a resistance equal to:
42— 1
vg-j-yX 8=705 kilograms per square centimeter.
The resistance of a cast-steel gun, allowing twenty kilograms per milli-
meter for the strength of this last metal, will be
42— 1
n_r^X 20=1760 kilograms.
Sixteen centimeter guns of cast iron reinforced loith steel lioops.
The guns of 16 centimeters are made from a single piece of cast iron,
forming the body of the gun, and a row of hoops of puddled steel ex-
tending to the trunnions. The tests to which the hoops were subjected,
before being received, insured the malleability of the metal but not its
elasticity. Thus the guns of this model which have been last fabricated
have less durability than their predecessors.
As a secondary cause of the inferiority of this model of gun, the
depth of the grooves must be considered ; this is somewhat lessened by
the form of a flat arch being given to the profile of the grooves so that
there are no re-entering angles.
21
The transverse diuieiisions of the gnu are as follows : exterior di-
ameter of the hoops 584 millimeters, diameter of the reinforce under
the hoops 472 millimeters, caliber 164.7 millimeters.
If we use these data iif the first example already given, and if taking
into consideration the want of elasticity of the hoops, and calling the
tensile strength 20 kilograms per square millimeter, the cast iron hav-
ing a tensile strength of 8 kilograms, we find 1360 kilograms per cent-
imeter for the resistance of the gun. These guns were constructed to
fire a shell of 31 kilograms, witli an ordinary charge of cannon powder
-3.5 kilograms, with a wad between the charge and the projectile. They
stand this fire, jbut in increasing the charge to 7.5 kilograms, with a pro-
jectile of 45 kilograms, the guns have always burst after a few rounds.
2. — Cast-iron gun^ lined icith steel tubes — Falliser system.
Let a'cast-iron gun pros ided with an interior steel tube be now con-
sidered.
As in this case the cast iron is not directly submitted to the same
causes of deterioration as exist in the preceding case, a tensile strength,
of 10 kilograms may be assumed. For inverse reasons the tensile strength
of the steel will be lowered.
Cast steel, which alone is employed for the manufacture of the central
tube, is often full of bubbles or air spaces. Hammering may cause these
flaws to disappear, but only in closing up the sides, so that the resist-
ance of the metal is not increased. It is possible by certain processes
of fabrication, such as casting the metal under a pressure, or using a
lieavy sinking head, to almost entirely avoid these causes of weakness ;
but it is necessary to avoid erring on the other side in attributing to
<;ast-steel tubes a less resistance than to hoops of puddled steel, which
are made from masses of metal relatively weaker.
For these several reasons, we will fix on 20 kilograms per square
millimeter as the tension to which cast-steel tubes should be submitted.
This strength of 20 kilograms corresponds, besides to the limit of
elasticity of low steel.
AVe will suppose, as in the preceding example, that the gun is a
^caliber and a half thick, and one caliber is of cast iron, and half a caliber
of steel
From the data given, Ei will be equal to 2Ro. and E^ to 4Eo; we shall
c-onsequently have
Pi=4^^i-2Xl0=6kil.
and
Po=^^(20+ 6) + 6=21^6
per square millimeter, or 2160 kilograms per square centimeter. If we
22
had fixed, as in the preceding example, 8 kilograms for the tensile?
strength of the cast iroD, we should have found:
Pi=4\3 and Po=18K90.
This value of Pq is about equal to that found in the case of the hooped
gun.
Thus regarding only the elastic tension, we recognize that the tubed
gun (Palliser system), for the same thickness of steel, presents about
the same resistance as the hooped gun. But the elastic tension is not
the only one to consider. The bore of the gun is liable to the accidents
of discharge, which may determine exaggerated pressures. In these cir-
cumstances cast iron, when forming the bore, soon reaches the degree of
tension which determines rupture, and cracks are formed which increase
the surface of gas pressure, and as the gun is again discliarged, sooner
or later the piece is destroyed.
Steel, on the contrary, employed under the same conditions, ^¥ill break
with difficulty under a pressure of 55 kilograms per square millimeter,,
and the limit of elasticity m^y be greatly exceeded without breaking j
thus the chances of accident will be much diminished. From these con-
siderations the tubed guns (Palliser) are to be preferred rather than
those of the hooped system.
3. — Guns hooped and tiibed.
Let us again suppose a cast-iron gun to have walls of a caliber and a-
half in thickness, of which half a caliber is cast iron, and in which this^
metal is placed between the exterior and interior surfaces.
The hoops are to be two-thirds of the radius of the bore in thickness^
and the central tube one-third of this same radius.
We shall then have :
Kow supposing that the tension of the cast iron is limited to 8 kilo-
grams per square millimeter, that of the hoops to 25 kilograms and
that of the tube to 20 kilograms, we shall have
122—102
2 — 122 -I- 102^" "^^^
from which
102—42
i*i=ro2i:4-2(S+^-5i)+4-5i=i^-^'5
Po=42~32(20-fl3.57) + 13.57=22.97.
Thus the gun will resist a pressure of 22'^97 per square millimeter or
of 2297 kilograms per square centimeter. It will consequently have
more resistance than the gun in the preceding example.
If we give a tensile strength of 10 kilograms per square millimeter to
the cast iron, we shall find for Po 2483 kilograms per square centimeter,,
and the advantage of the mode of construction in whicli tubing is com-
bined with hoopifig becomes more evident.
>4
•23
§ 4.— Conditions of the iiia\iiuiini ol* resi^iitance.
Ill the preceding examples we have arbitrarily given the thickness
of the two or three concentric tubes, which constitute the reinforce of a
gun, taking as a guide the practical considerations drawn from the na-
ture of the metals employed. Xotably. in relation to cast iron, the fol-
lowing conclusions have been reached :
1. That in a gun of this metal, simply strengthened by the addition
of hoops or of a steel tube, the mass of steel should be but a small i>art
of the whole mass of the piece, or the economy which is to be kept in
view by the combined employment of the two metals will be lost.
2. That cast iron, on account of its brittleness. should only be used in
large masses of considerable thickness.
3. That this thickness is necessary in order that the gun may resist
transverse strains proceeding from the exterior or resulting from the
discharge.
4. Finally, it is necessary to increase the resistance which the circu-
lar part of the body of the gun opposes to the pressure of the gas at the
bottom of the bore, since we are ignorant how mucli the compressive
energy of the metal in the direction of the radius will act upon the re-
sistance. It would not be without interest to know, theoretically, the
division of the reinforce which will give the greatest strength to the.
gnn.
And in general to know what relations of this kind it is necessary r* •
establish between the different elementary tubes in order that the giin
may attain its maximum resistance, having given the number, the
nature of the elementary tubes, their order of superposition, as well as
the exterior and interior radiiof the built-up tube itselt^ let us imagine
then a tube composed of any number of simple concentric tubes, and
consider two of the latter occupying any position whatever, but follow-
ing one the other immediately in the order of superposition. Let us
call:
i\ the interior radius uf the smaller of the two elementury or
simple tubes,
R. the exterior radius of the greater,
p. the radius of the surface of junction of the two tubes.
P. the normal pressure on the surface whose radius is r.
P'. the pressure exerted on the surface whose radius is K.
T, the limit of elasticity of the interior tube,
T', that of the exterior tube.
Let it be admitted that all the radii remain constctnt. witli the exce^)-
tion of ,o, which we will consider as variable. The pressure P is repre-
sented by the formula:
(See formula (5).)
24
V is coustant because it depends solely on the limit of elasticity and
the radii of the elementaiy tubes, which are exterior to the two under
consideration, and by the hyi^othesis these quantities do not vary. R
and r are given, and P is the function of the only variable, p.
The pressure Po, which measures the resistance of the gun, augments
and diminishes with P, and only with P, while the limits of elasticity
and the radii of the tubes which are interior to the radius r remain con-
stant; Po will only be a maximum when P is at its maximum. We will
now seek the value of />, for which P is a maximum.
Reducing the terms of P to the same denominator and adding them,
we have
' p-/ ( T-2T0 + />^[(B^-r'^) T + 2B^T^ + 4R^ P^] -W r' T
The differential of P after reduction gives the following equation :
P^ [2W P' + (2R2 + r2) T' - r2 T] - 2/>2 R2 r^ (T - T')
- R^ r2 (2P' -f T + T') = 0.
Putting
we find
2R2 P ^+ (2R2 + r^) T^ — r2 T = A,
K2r2(T — TO = B,
R4f2(2p/4-T + T')=C,
^ A
C is always positive, since P' is a quantity essentially positive and
T and T' represent positive tensions. If A is equally positive, the prod-
uct AC will be positive, and the radical will give the sign to the value
of ^'^5 //- cannot then be negative, and the radical is to be taken with
the sign +.
If A is negative, B will give its sign to the numerator of p'^] but B
will in this case necessarily be positive, and A cannot be negative un-
less T is greater than T'; p^ will then be negative and ^o imaginary;
then, if there is a maximum, A is positive, and from what has preceded
the radical should receive the sign +.
Thus in all the cases in which there exists a value of p which renders
P a maximum the square of /> is given by the formula:
(7) P'=
B+VB'+AO
If the two tubes under consideration have the same limit of elasticity,
whivih occurs in treating of two rows of hoops or two superposed steel
25
Tubes, formula (7) becomes more simple since T = T^ which introduced
in the values of A. B and C, gives
s o- = llr.
If there are more than two consecutive elementary tubes, having the
same limit of elasticity, any two of them whatever, if next to each
other, should satisfy formula (8), from whence the following practical
rule :
In a tube built uj) comecutively with elementary tubes of the same metal^
or icith tubes of different metals but having the same elastic resistance, the
raflii of the surfaces of contact of the elementary tubes should increase in
{fco metrical jjrot/ress ion from the interior of the smallest to the e.rterior of
the largest tube.
If a built-up tube is formed from a series of simple concentric tubes
oi the same metal, of which the radii have been determined by this
rule, the calculation of the resistance becomes easy and reduces to the
summation of a geometrical progression. 'Formula (5).l
If we wish to find the values of p. which will make a maximum, Urst
of the sum of tlje first two terms of P, second the third term, formula
'6 1, it will suffice to introduce successively in A. B and C the following
hypothesis: first. P' = 0: second. T=0. and T'=0.
It will be found that the values of o^, which correspond to the two
maxima, are. for the sum of the two first terms of P.
(9) . . . ,- = "'
-and for the third term.
T_T/^+E-/V^T'[E-(T-hT^)^+r^(T -T :
Frequently, with a sufficiently close approximation, we may neglect
,-^(T'— T) by the side of Pv-(T' + Ti and 2R-T\ and replace T + T^ by 2T':
we shall then have for the two first terms as for the third.
y- = ~Rr.
Thus the sum of the two first terms of P formnla (G) often becomes a
maximum for a value of .o~, which difiers little from the product Pr,
while the maximum of the third term always exactly corresponds to
o-=Jir. This circumstance gives, then, a greater practical utility to
the rule already given, and often permir> irs extension even to element-
ary tubes of ditferent elasticities.
Eeturning to the case when it is wished to determine exactly the re-
lations of thickness giving the maximum resistance.
Formula (0) applies to the case of a tube composed only of two ele-
mentary tubes of different metals. If there are more than two element-
ary tubes, or if of difierent metals, or if they have not the same limit
of elasticity, it will be necessary to resort to formula (7), and to give to
26
the elementaiy tubes arbitrary thicknesses, which can afterwards be
corrected by successive substitutions. (See third example, further on.)
A division of the tube, established from the rule already given, may
be taken for the calculation, this rule giving, as has just been exi)lained^
a first approximation, even when the limits of elasticity of the metals
employed differ notably among themselves.
t
§ 5. Practical examples.
1. — Hooped cast-iron guns.
Let us seek the interior radius of the hoops corresponding to the aiax-
imum of resistance of cast-iron guns hooped with steel, and suppose that
the total thickness of the reinforce is a caliber and a half. We will fix
the tensile strength of the cast iron at 8 kilograms and that of the hoops
at 25 kilograms per square millimeter.
Making, in formula (9),
T=8, T'=25, R=4r,
we shall find
hence
,o^=2.9r%-
p = 1.7r.
From this we deduce the thickness of the cast iron
1.7/-_r= 0.7r
and that of the steel
4r — 1.7r=2.3r.
The value of P, or the interior pressure measuring the resistance of the
gun will be raised to 3270 kilograms per square centimeter.
But the thin cast-iron tube will not practically fulfill these conditions.
It is not possible, then, in this particular case to admit the relation of
thickness between the cast iron and the steel, which would oppose ttie
maximum of resistance to longitudinal fracture; but we arrive at the
same time at this curious result, that the resistance of a cannon would
be augmented in giving it an interior liniug of about 0.7 the radius of
the bore.
2. — Cast-iron guns icUh steel tubes.
Let us pass to the opposite system of construction, that of a cast-ii'on
gun lined with a steel tube, and having, as in the preceding case, a
caliber and a half of thickness. Bearing in mind what has already been
said on the nature of the two metals, we will take as the limit of tensile
27
strength 20 kilograms per square millimeter for tlie steel tube and lCt>
kilograms for the bodj^ of the cast-iron gun.
In formula (9) making
T=20, T' = 10, E=4r,
^•^=5.52 r2;
we find
whence
P=2.35r.
The thickness of the steel will be
p — r=1.35r,
that of the cast iron
E-,o = 1.65r
and the pressure, P or Po=2210 kilograms per square centimeter.
Here the thickness of a cast-iron tube, joined to its large diameter^
will give sufticient resistance for all causes of destruction that may
occur, and which are partly supported by the central tube. But it may-
be remarked that the maximum pressure that we have just found, 2210
kilograms, differs very little from that of 2160 already obtained in a pre-
vious example, by fixing a priori the thickness of tlie steel tube at half a
caliber ; from whence it results that t.his last thickness is to be preferred,,
since it permits the realization of a certain economy of construction, and
of notably increasing the envelope of cast iron.
3. — (thhs^ both hooped and tubed.
Let us consider a cast-iron gun strengthened by an exterior row of
puddled-steel hoops, and with an interior tube of cast steel, and deter-
mine the iuterior radius of the hoops and the exterior radius of the-
central tube, so that the maximum resistance may be realized.
We will supjjose a total thickness of one and a half calibers, the steel
to have a tensile strength of 20 kilograms for the cast metal, and for 25
kilograms for the hoops; the cast iron to have a tensile strength of 10
kilograms. In this case, as there are more than two tubes, foruuila 7
must be employed.
Let the reinforce at first be divided as though the elementary tubes
were of the same metal, and let the radius of the bore be taken as^
unity.
From the method of division adopted as a point of departure, Ki, ex-
terior radiu-s of the central tube, will be equal to >yi = 1.59. Starting^
with this number, we can determine the A^alues which approximate more
and more closely to this radius Ej and to E2, the interior radius of tlie
hoops.
28
First operation. — Calculation of Bi.
The given values are :
r =Ei= 1.59, whence r"^ = 2.53,
E =E.>= 4, whence E2 = 16,
P'=P3= 0,
T = Ti = 10,
T' = T, = 25.
^Substituting in the values for A, B and C, we find
A = 838, B = -607, = 22.680.
Using these vahies of A, B and 0, in formula (7), we have
^/2 — 4.55^ whence p = E2 = 2.13.
Second operation. — Calculation of Ri,
The given values are :
r^ = E% = l,
E^ = E^ =4.55,
T =To =20,
*
T' = Ti = 10,
Introducing as before in the values of A, B and C, Ave have
whence,
A = 207, B = 45.5, C = 1.196,
;r.2=2.65 and /. = Ei = 1.63.
Third operation. — Calculation ofR^-
r2 =
2.65, E^ = 16, T = 10, T' = 25, P^ = 0,
whence.
A = 839.5, B = - (jm, C = 2.373
:and by formula (7),
^,2 = 4.71, . ^o=:E, = 2.13.
Falling back on the value of E2, already- found to within nearly 0.01,
the values of Ei and of E2 will consequently be determined with a
sufficient degree of approximation.
Thus the resistance of a gun will be carried to its maximum when
radius Ei = 1.63 Eo, and radius E2 = 2.13 Eq. The thickness of the three
•elementary tubes will then be :
the central steel tube 0.63 Eo,
cast-iron body 0.50 Eo,
the hoops 1.87 Eq.
29
The resistance of tlie gun, or Po, will be 4.")4:!) kilograms per s(iiiare
ceiitiineter.
A constructiou like this, however, is not admissible, since the cast-
iron body would not be as thick as the central tube, while the hoops
would have such dimensions as would require them to be regarded as^
the principal element of the gun.
4. — Malleable tube irifhout reshtance plaeed in a resistine/ (jun.
AVe may conceive of a homogenous gun that should receive a central
tube oidy designed to transmit pressure. It is ])roposed to discuss the
relation between tlie resistance of the' [)iece and the thickness of this-
interior lining.
The tube may possess an elastic comi)ressibility vsuflicient to support
the compression due to the initial tension of its envelope, but without
l)Ower to resist the least eftbrt of tension; or rather we may imagine^
that it possesses extensibility and malleability in an unlimited degree,
that it follows all the movements of expansion and contraction in the
body of the cannon, but without opposing the slightest reaction.
In these two cases we shall have the value of //, which corresponds to
the maximum of resistance, by making in formula (9) T=0.
If it is granted further that R be made equal to 4r, we shall find
^o = 1.53r.
Consequent!}' the thickness of the tube will be 0.53r, and the value of
Po=1.04 T'.
Thus, by the addition of a central malleable tube transmitting the press-
ure of gas to the resisting walls, the gun acquires a greater resistance
than if its thickness had been unlimited, and the thickness necessary
to give such a tube would be about one-quarter of a caliber.
o. — Cast-iron or steel guns with bronze tubes.
Bronze may be employed as the malleable metal in the illustration just
given, but this metal having a resistance of its own, it will be necessary
to take it into account.
Let us imagine a cast-iron gun with an interior bronze tube, and see
what the thickness of this tube must be, so that the resistance shall be
the greatest possible : let us sui^pose, besides,
E = 4r.
Bronze may be considered either as elastic or malleable. In the first
case it should not be submitted to a tension greater than 2 kilograms per
square millimeter ; in the second case the use of old bronze ordnance
shows that it supports heavy strains under the same dimensions, as
well as cast iron, and that it can withstand a tension of 10 kilograms.
30
In the first case, considering the use of bronze within its limit of elas-
t:icity, if we make
T=2andT'=10,
formula (9) gives
/o=1.65 r.
The thickness of the tube will then be
O.Go. /■;
and taking for the calculation of resistance
Eo=r, Ei=/>=1.65r, E2=E=4r, T=2 kil., T'=10 kil.
>
we find 1124 kilograms for the resistance of the gun.
If, considering the bronze as a malleable metal, we make
T=:10 and T^=10,
formula (S) will give
The thickness of the tube should be taken at half a caliber. The cal-
culation of the resistance will give 1560 kilograms per square centimeter.
We see that these results are verj* inferior to those obtained from
^teel hoops or tubes.
If, instead of a cast-iron envelope, we conceive of a steel envelope, we
^hall find
1.36r
for the thickness to be given to a bronze tube, and
2535 kilograms
for the resistance of the piece.
Dutch system of conversion.
The system of construction which consists in strengthening cast-iron
guns with bronze tubes has been realized in Holland with good results.
The old guns were reamed out and filled in with bronze castings,wliicli
were bored out to the proper caliber.
6. — Bronze guns ivith steel tubes.
It must be supposed in this case that the steel tube possesses a cer-
tain elasticity. It is necessary that the bronze should not be strained
above its limit of elasticity, because if it were it would separate from
the interior tube when the latter contracted after the explosion. Con-
sequently, in formula (9) we will put
T=20 and T'=2.
If we again suppose
E=4r,
we shall find
P=4rMr',
31
that is to say, a value of p greater tbau R ; which signifies that there is
no value of p which gives a maximum, and that the gun should be con-
structed entirely of steel.
The combination which we have just examined is not, then, a suitable
one as far as the resistance is concerned. It was nevertheless attempted
in the construction of a gun at Gavre; the promjit destruction, how-
evar, of the piece, of which the tube was of good Krupp steel, confirmed
the indications of the theory.
Exx)eriments have lately been renewed in regard to tubing bronze
guns. The end to be attained was to give greater hardness to the sur-
face of the bore, and to jDrolong by this means the life of the piece.
The difference between these new efforts and former attempts consists*
III the substitution of steel for iron. It is important that the steel tube
sliould be malleable and preserve permanently the extension which it
acquires under the pressure of the powder-gas, otherwise it will sepa-
rate from the bronze.
C(MA.
CJLrx/^n^
iLc Ou/muj^ ^' ^
1
?•
.'H.I
C<iC -
Longitudinal and Hoop Tension in a Thick Hollow Cylinder.
By Lieut. C. A. Stone, U. S. N.
Consider a thick hollow cylinder, closed at both ends and pressed from
within by a pressure P, and from without by a pressure P' . , The notation
being the same as that given on page 6 of Virgile, we have, for a longitudi-
nal section through the axis:
tdp = PR — P' R\ (1)
R
for a cross-section, we have
•^' ^ PR'—P'R'^ ,^.
spdp = r —^ (2)
R ^
where s is the longitudinal tension.
Let t =f' [p): then from (1),
t=:f'{p)=-p-p^^, m
dp
Let s = — ; then from (2) ,
/;
. we
/;
From (3) and (4),
t
and 2s — t=:—p. (6)
dt)
First: let f = s; then from (5) -^ = ; .^ — p = m, m being a cc
From (3) and (4). we have also, ^ =sz=??i. This satisfies (1) and (2).
For the second sol
(3) and (4) , we have
^ '- 2- dp' ^'^
dt)
First: let f = s; then from (5) -^ = 0\ r. —p=^m. m being a constant-,
s = m. This satisfies (
For the second solution, let f = — s in (5), then t =z -7-" and. from
4 dp
3p dp dp 4 dp
(7)
"^ 2* dp' • • p ~ 3' p ^
from which we see by integration that p is of the form ~^.
Differentiating, we have,
dp ^ _a
'^~~ 3-~|'
whence t = -L.^^^^. (8)
4 dp 3^^
We have also P = — -- and P' = -^ri • (9 )
Rj -K 3
Substituting these values in (1), we have
R'
%=PR-P'R' = arj: L_-|,
(10)
RP'
which is thus satisfied.
Combining the partial equations, we have,
t = m + -^, (11)
op's
a
(12)
p — —m^ J-; (13)
pz
.\ St--p = 4:m, whence 3 T— P=i^T' — P' (14)
Act, 4 4
and ^+i>=:~^, whence (r 4- -P)i^' = (^' +P')i^'^ (15)
Sp-i
P P'
Eliminating T' = T — -f- -^, we have
o o
S E^ {T -^ P) =^B'^ [Q T -- P + 4: P')
••• T=-^ 4 4^ • 16)
From (12) nnd (13), we have
s-|-p = -^, whence {P+S) E^ z=z {P' -\- S' ) R'^ (17)
and 3 s H-_p — 2m, whence 3 Sf -j- i? = 3 S'' + P', (18)
4 4 / P 2 P' \
whence P"^ (p_^ g) = P'^ V'^' H" ^ "^ "T^j '
^.^^^P(3Pf-P-t)-2P-pl
3(P'|-Pf) • ^
Where S, and 5^' are values of s for p=:R and p-=z R\
From (17) and (18), we have
^l(s+p)=:pt(.g + P)
and 3s + P = BSf-hP
.♦. p3 (3^_2.s-}-P)=P^^P-j-Sr)
^^ S (3 p^ — P^) -h P (p*— P^) = 2 s p^
— ^ ^^ ^ ~ -^!}jf.:P_(/^^— -^^)
Substituting the value of aS' from (19), and reducing, we have
P E^ (3 p^ — R' h — P' R'~hsp'^—E^)
ds __ 4PJ R'Up—P')
whence -j ^^ ~ —-
^'^ 9p3 (P'3_^/f)
and R' ^ R f ^ ^® ~^' therefore the longitudinal tension is greatest on the
outside.
Let p = R' in (20)
2PP3— P'(3P'3~P3 )
S":
(21)
3(P'-^— P«)
From (14) and (15), we have
3^— p=:3r— P
and p^ {t-\-p)=zR^{T-j-P)
4 4 4 4
r (3 o"^4-p3) — p (p'^ — pn
whence t= ^' ' ^ ^ . (22)
4p3
Substituting the value of Tfrom (16), we have, after reduction,
P R^ (3 p^ + R'h - P^ ^^ (3 P^+R"^ ) .._.
3p3 (^'3 _^3)
Differentiating
^^ /^ 3~ is negative; therefore the greatest hoop-tension is on the in-
If P > P'
and R >
side, where p= P; and its value T is given by equation (16).
To determine m: from (14)
substituting the value of Tirom (16), and reducing
PR^ — P' R'^
m = 1 ^— • (24)
R'^—R^
To determine a: from (11), after substitution of particular values
a = dR^{T—m)
R^ R'^ [P—P')
whence a=z ^ ^ , (25)
R'^ —R^
Example :
1 r ,7. 19
Let P = 1 j I 45 ^
P' = 8 . Qvi/i Tvro ViC^^cx ! 5^'= — P
1 and we have \^ — 77
1 13
l^ = -45^
and P' =:
»S being a longitudinal compression, since it is negative. If the longitudinal
tension were uniform over the area of the cross -section, its value would be
P
63-
From equation (20). we see that, when P^ = and Sp^ =:P^%orp = .489P^
about. 5 = 0; which shows the position of what may be called the neutral
surface, within whic^there is a longitudinal compression, and without which
a longitudinal tension. In (19), when 3 P^ < P^^^ g jg negative, showing a
longitudinal compression. Making P^ = P in (20), we find the longitudinal
tension s is uniform and = — P, as might be expected.
From (20), making s = 0, we find, for the position of the neutral surface,
p^ =: 4 4'
3(PP^^— P^P^3
From (12), when s = 0, p^ = -— — , which agrees with the above.
Formula (16), giving the value of the maximum hoop-tension, differs con-
siderably from that heretofore used, in the deduction of which the longitu-
dinal stress was considered zero or constant. The existence of a neutral
surface of longitudinal stress is of great interest in the construction of built-
up guns. That a longitudinal contraction may accompany a circumferential
expansion, and must do so under certain circumstances, is a familiar result
of experience.
The above is to be found in Vol. VIII, No. 3, of the Proceedings of the U.
S. Naval Institute ; the notation having been so changed as to make it the
same as that used in Virgile's formulae.
SHRINKAGE IN THE TUBES OF A BUILT-UP GUN.
Up to this point, we have assumed, in all calculations, that each simple
tube resists at its elastic limit at the moment when the pressure in the pow-
der chamber is greatest. In order that this shall be so, it is necessary that
the gun shall be so built that when immersed in any medium, as in air, the
simple tubes shall be in a state of initial strain; the inner tubes being com-
pressed, while those near the outer surface of the gun are stretched. It is
our object now to determine the amount of this strain.
<;
In the figure, let the circle of radius A represent a gun-cylinder com
posed of a number of simple tubes, upon which one or more simple tubes are
yet to be placed; let O C be the inner radius of a tube to be placed upon the
first: the quantity A C ■= O A — OC, called the shrinkage, is what we are to
determine. Now, when the outer tube is expanded by heat, slipped over the
inner one, and the system is allowed to cool, the surface of contact will have
some radius O D ; greater than O C. and less than O A. If more tubes are
similarly put on outside, the radius of contact will become still less, as O E.
If now, the powder pressure be generated, the radius will increase, as to O F.
In each of these states, the value of the shrinkage is the same, being, in
the last, equal to the sum of A F and F C. A F is the amount by which our
first gun-cylinder would increase in exterior diameter if, at the same instant,
it were freed from the powder pressure within and the pressure of the tubes
outside it. F C is the amount the outside tube would contract in diameter
if the pressures on its two surfaces were similarly made equal.
These quantities are, respectively, the first and second terms in the second
member of equation (7).
P
By means of Hooke'a Law^ expressed in our notation i = ^, we may, hav-
ing tabulated values of E, determine i froin p, or vice versa. The value of
p in the present case requires further investigation, since the stress at any
point in the mass of the gun is not confined to a plane.
If a rectangular paraiielopiped, of edges a, 5, c, be subjected to stresses
parallel to its three edges, its dimensions will be changed. It is clear that,
within the limits within which the alterations of dimensions are proportion-
al to the forces causing them, or within which Hooke's Law is applicable,
the effect of any stress is independent of the existence of prior stress. This
being admitted, if we denote by i, the elongation produced in the direction
of a by the force X, and by e, the contraction in the two directions at right
angles to a caused by the same force; andby ^2, e^ and ^^, e^ the same quan-
tities caused by forces Yand Z, parallel to band c respectively, we shiill
have
^^=^-^-^8 1
The ratio of e to i is generally admitted to be constant for the same sub-
stance within the limits of stress considered; its numerical value will be taken
as i, that being the best determination. We may then write,
Ac= L\z-k(x-hY)^ I
Replacing X, — Y. and Z by the quantities T, P. S, in the notation already
in use, we have for any change in the diameter of a tube subjected to these
forces,
E
We shall in all cases neglect S; the force parallel to the axis of the gun.
Thelettersi2^,i^^,i^^, .P^.P^.P,. , T^,T^,T^, , T\, T\.
T' will have the same meaning as hitherto; and we shall call the
moduli of the plasticity of the various tubes E^. E^,E,^
rp ^ rp ^ rp ^ ^ bciug the limits of elasticity of the metals of the various
tubes, aie known, P^, P^, P^, can be calculated by formula (4), of the
translation of Virgile; and T\, T\, T ,. may be found from the law that the
difference of the hoop -tension and pressure at any point is a constant, for ex-
ample, I _
Consider now, the case of a gun consisting of two simple tubes which are
each stretched to their elastic limit, and thus doing their best work, when
they have the powder gas, with a known pressure, within, and zero pressure
without; suppose that the powder pressure is removed, and. at the same in-
stant, each tube is freed from the other: the outside diameter of the inner
tube would change per linear unit by the amount
and the inside diameter of the outer tube would change, per linear unit,
Therefore, the shrinkage will be,
Calculation of the Shrinkage at the Various Stages of Construction.
If the simple tubes were, in practice, each fitted separately, and then put
together, the method given of calculating the shrinkage would suffice; the
exterior of each tube, after it has been put in place, is however turned and
finished before adding another tube, and thus the determination of the shrink-
age of this tube becomes more complex. ....
Suppose a tube is in equilibrium under the internal pressure it is intended
to support, and with its internal surface at the elastic limit, and that this
condition is, in any manner, modiiied; the pressures and tensions will un-
dergo variations which we will represent respectively by Po.Pi, P21 P "
p . ^; ^0' ^11 ^2 i'- t'oj ^17 h^ ^ -
We shall establish the relations between to and p^, the variations caused
by the supposed change at the surface of the bore; and f'„ _ i,f,, andp,
vvhich occur at the surface of contact of any two tubes: we shall thus be able
to determine the shrinkage which should be given to the second of the two
tubes.
The equilibrium of tlie (n + 1) ' tube, in the first state, is given by the
equations,
T. — P„ = r — P„4.i;
the equilibrium of the same tube in the second state, by the equations,
[{T. + t) + (P,. +p.) Jp;= [(r; 4- O + (^" + 1 +P' + ^0^'h- 1
[T. + 1) - (P. 4-p.) = [r + t) - (P„ ^ ^ +p_ ^^);
from these four equations, we derive,
two relations which might have been anticipated.
Solving these equations for t' and p . , we have
P' + p' e\ — p'
t\ = t, ^^ 3 p,.-^^ -; (1)
these will give the values of V andp when t, andp, are known.
""T 1
The exterior diameter of the {n -\- 1)" tube, in its passage from the first to
the second state, will undergo a change, per linear unit, of
the interior diameter of the [n -\-2)' tube will, since the surfaces of the two
tubes are in eontact, undergo the same change; we have therefor,
ii/,. \ " '.-1-1/ U_^i\ "+i »+i/
whence
If now, we put n — in (1) and (2), we have
to =Ao to -i-BoPo,
Pi = C^ to+D^po:
Ao, Bo, Ci, Di, being known. Putting n =: in (3), and substituting the
values of ^0 andp, just found, we derive from (3),
t, = G,to-hS,po.
Similarly, making n = 1 in (1), (2), and (3), and using the values of #1, Pi
ti, and P2, we derive equations of the form
ti ^^1^0 + -^iPoi
P2=C2to +D2P0,
^2 = Gr2 to -^ ^2^0)
where the cofBcients of to andpo are known.
We have, therefore, generally,
/ A t -\-B p , (4)
'.-1 "-1 «-l
p.. = ato-hD,.Po. (5)
t„ = G..to-\-H..Po' (6)
If now, in building a gun, the first n tubes are in place, and we wish to
determine the shrinkage to be given the {n + 1) ' tube, we conceive the gun
to be first in equilibrium with the {n -\- 1) ' tube on, with the powder pressure
within, and all the tubes at their elastic limit; we then, simultaneously, re-
move the powder pressure, and free the n tubes from the {n + 1) '. The
change in the exterior diameter, per linear unit, of the system of n tubes
will be
to:-.+i..)
that of the interior diameter of the [n + 1)" tube
Calling A the shrinkage per linear unit, we have therefore,
A.
1 ((_+tp„)+^(r, + iP.). (7
"1
Now, to represent analytically the passage of the system of {n -\- 1) tubes,
from the first state, that of resisting together at their elastic limit the powder
pressure, to the second, that without the powder pressure and of separation
at the n"' surface of contact, we havepo = — -^oi and p, = — P„m (4), (5),
and (7). Thus (7) becomes,
A„:=^ A' -iP„)+-^f r„-f iP„V (8)
to compute f'_^, we find to^ by substitution of the known values of p^ and p^
in (5), and this value, v/ith thatof Poi ^^ (4), gives f_i; which finally, in (8)'
gives A.
In the case in which the tubes are of the same metal, we have EZ^ = E,.
= £^-, whence, from (8),
A„
iOL + ^") (^)
also we have
whence
(10)
2po Ro-pAR -\-Ro^
-K„ — -Ko
if, in this equation, we make Pq-=z — P^ and p„ = — P„, we find
Tangential Compression at the Surface of the Bore,
When a built-up gun is not subjected to powder pressure within, the inside
surface is compressed; and, as the amount of this compression might become
so great as to injure the metal at this point, it is important to ascertain iis
value in any given case. In the case of sudden alterations of pressure, vibra-
tion would be set up, and the danger of injury would be increased as shown
in the determination of the effect of a sudden force.
If the gun in question consists of n tubes, we have, from (5), by putting
p.. =: 0. and Po = — Po
whence
t, = '^. (12)
In this equation, to is the negative change of hoop-tension which the sur-
face of the bore undergoes in passing from the extreme tension to the extreme
compression. T^, being the first of these limits, the second will be represen-
ted by the algebraic sum Tq -p ^o- This is a negative quantity whose abso-
lute value is the compression per linear unit.
If all the tubes have the same elasticity, we have, from equations (10),
«„=-P„^i±^ (13)
Rn JRo
Examples.
Cast-iron Gun icith Hoops of Steel.
Suppose the gun has one caliber's thickness of cast-iron, and one-half cali-
ber's thickness of steel outside; the elastic limits of the cast-iron and steel
are 8 kilos and 25 kilos per square millimeter; the moduli of the two metals
are 10.000 and 20,000 kilos respectively. As shown on page 20 of the trans-
lation of Virgile, Po« is 19 kilos; and Pj is 7 kilos per square millimeter:
hence
r; = ro-PoH-Pi=-4.
The change, per linear unit, of the exterior diameter of the cast-iron tube
will therefore be
~ t"tji — — .000 166,
10.000
that of the interior diameter of the steel hoop
25-h ^
= -h .00136-
20.000
The sum of these two quantities, or .001533, will be the shrinkage required
per linear unit.
To find the compression of the inner surface of the bore. The cofficient of
fo and_Poi i^ (1) ^^^ ("^l ^^e
2 PI
— 9
r\-rI
4
2Rl
— y
We have
therefore, from
(1) and (2)
t=lto-
tPo.
Pi=-tt
3 + e.Po-
Since,
in
the
case considered, — "— — =
we have,
from
(3)
^.=2/-;-+
ipl^
but
(2/o = V^o-fPo
hp:=-
-^l^^o+^
-%~
Po
hence
t,=^¥'
-to-H'Pc
•
Putting,
in
(2),
vi=l, Pi =
3, p2 = 4; as indicated by the data of
the
prob
lem. we
find
P2=— 3^2
^i+!fPi
lint
j-sV^:
= -fHUo
+
mhPo
UUt
I IfjPl
==-^hto
+
HmPo
hence
^^ W. 0: H 9 / I 1 (i 9 ■ .J I- ^.
p.j, !. -v 3 - t'O n^ i' A- .; J 2-^0 •
In this equation, we make p2 = 0, and 2>i, '— — 19; whence we find
#0— — 20.0.
The tension of the inner surface of the bore when the internal and external
pressures are the same is therefore
f + 'i'o = — 20 -f 8 — — 13.
This is the force of compression in kilos per square millimeter at the sur-
face of the bore. It is well within the strength of the metal..
This example can be otherwise worked out by finding the value of ^o from
(12). and then using (4) with the value of f „ just found to find t ^, we finally
substitute the last in (7) to find A, In the first operation it is to be noticed
that the value of n for which p= is n =: 2. (12) therefore becomes
The coefficients are easily found, as indicated in deriving (4), (5), and (6);
and F(. is the internal pressure.
For a second example we will take a gun made of two steel tubes and a
row of steel hoops: the walls being one caliber thick, and the successive radii
increasing in geometrical proportion. The modulus of elasticity is the same
for all; namely 20,000 kilos per square millimeter, and the tubes are to be set
to pull at 20 kilos and the hoops ai 25 kilos per square millimeter at the mo-
ment of maximum tension.
Making i?n - 1; we find J?, = 1 .44. F, = 2.07. and R. — 3. We shall also
find P,_ = 8.72. P, :=--: 18.73. and P., ^ 32.24.
For the shrinkage:
In formulae (11) and (9i. making successively n -.=- 1 and ii =2. we find:
t'^ = — 6.49 and A^ = .000673.
t'^s=z — o.Q2 and -\ , = . 000969 ;
the two tubes should therefore be adjusted with .00068 shrinkage per unit of
length of diameter, and the hoops with .00097. If the gun were of 6 inch
caliber, we find i2„ = 3. _R, — 4.3. Zy. — 6.2 jR,--9: for the difierence of
diameter at the surface of contact of the two tubes. .003 inch and at the other
surface .006 inch
Compression of bore:
Making n=zS in (13). we find
fo= — 40.3.
whence T,, -\~t^ = — 20.d.
Thus, in round numbers, we vary the tension of the metal at this point
from -f-20 to — 20 kilos per square millimeter, or from -i- 28.446 to — 28.446
lbs. per square inch (1 kilo per square millimeter = 1422.3 lbs. per square
inch). The most convenient way of reducing this dangerous excess >vould
be to decrease the initial strain of the steel hooj).
'/:
ON THE STRENGTH OF A THICK HOLLOW SPHERE.
Let the figure on page 6 of Virgile represent a diametral section of a hol-
low sphere, the notation being the same as given there. The pressure to be
resisted at the section is
^{PR^—P'R'^);
and if the section of the metal be conceived to be divided into an indefinite
number of concentric rings, the breadth of one of these rings being dp, it
appears that the total resistance of the section will be
2 TT I tp dp\
J R
and hence the equation to be fulfilled will be
*R'
tpdpz=PR^-P' R'^ (1>
R
Let t = - — ' in (1) ; and we have, after integration
2f{R') — 2f{R) = PR^—P'R'^',
whence, since R' and R are independent of each other, in general
2f{p)=-pp\
and 2f{p)=—2pp—p^-^-=:2tp (2).
As in the case of a thick hollow cylinder, the solution is effected by two par-
tial solutions. From symmetry it appears that the axes of stress at any par-
ticle must be. one in the direction of a radius, with the pressure p along it,
and the other two in any two directions perpendicular to the first and to
each other, with equal tensions t along them.
For the first case, being that of a fiuid tension^ we put — p = m: whence
dp
we find, from (2), - - =r O, and t = — p. Substituting these values in equa-
tion (1), reduces both members of that equation to
m{R'^-—R^).
For the second case, being that of a pair of circumferential tensions, each
equal to half of the radial pressure, we put_p = 2 ^ in (2), whence
.__£_ dp
*- 6- >""• ^^^
T 4. dp 6a ...
Let — — = —, (4)
dp p'
then substituting in (3), we have
Substituting the values of t andp from this equation in (1), we find, after
integrating.
This equation gives c = 4;
From (5) then, we see that the second case gives .
t — — — ^
P^ ~ 2'
Combining the two solutions we have,
^=^-f-m| (6).
2a \
whence {t-\-p) p^ = ?>a\ (7)
2t
9) p3_3al
p=:3m )
The constants a and w maybe determined from (4) and (6). Their values
are
_ {P — P')R' E'^
PR^ — P'R'^
m
R'^—R^
From the first of (6), we derive
■R' \ // 2(P4-T)
=V(^
R " ^ \2TH-3P'— P/ ^^^
Since P', in shells, is very small when compared to P, this equation shows
us that, if P is equal to, or greater than 2 jT, no thickness of metal will be
sufficient to withstand the pressure. We may also solve this equation for
P, thus obtaining a formula which would enable us to proportion blowing
charges in shells.
The above result is taken from Professor Rankine's Applied Mechanics.
GUN DESIGNING.
A gun. like all other machines, must be designed to fulfil certain definite
conditions. Its projectile may be required to pierce a given thickness of
armor at a given distance; or weight of piece maj' be the limit, and it may be
wished to throw the most powerful shell or shrapnel to a given distance with
a given elevation, consistent with that limit. To work out problems of this
nature, it is all-important to possess an accurate knowledge of the action of
the charge inside the bore. By means of the Noble chronoscope and the
crusher gauge, this knowledge is obtained, and w^e shall now explain how^
the indications of these instruments are employed to assist in determining
the proportions of ordnance.
Gunpowder is not properly so much an explosive as a substance burning
and giving off gas with great rapidity. It offers in this respect a marked
contrast to gun-cotton, dynamite, and other true explosives. If one of these
agents be detonated, the detonation is immediately carried through the mass,
whatever its size, and the whole at once turns into gas. Gunpowder, on the
other hand, as far as is known at present, cannot be detonated, but simply
evolves its gas by burning in layers from the outside to the inside. Thus a
large grain will take longer to burn up and become entirely converted into
gas than a small one w411; hence the effect of enlarging the grains is to render
the action of the charge less violent, the composition of the powder being the
same in all gun charges. The projectile is driven out of the bore by the pres-
sure of the gas on its base — that is. on an area that varies with the square of
the calibre. The weight of projectiles of similar form varies with the cube
of the calibre. Hence the larger the gun the heavier will be the column of
metal or projectile driven by each square inch of its base; and the great'er
must be either the pressure applieJ, or the time of its application, if a given
velocity is to be attained. The great object of the gunmaker is to obtain the
highest possible ratio of muzzle velocity to breech pressure. His ideal would
be a charge so arranged that a pressure equal to the amount the gun is con-
structed to bear should be uniformly maintained till the shot has left the
muzzle. Science is still a long way from this, but has done a good deal towards
it in the last few year^. A charge of gunpowder, composed of service ingre-
dients, in service proportions, exploded in a closed vessel at a density of 1.00
(equal to that of water), sets up a pressure of 43 tons per square inch: at a
density of 0.75. of 23.2 tons; at 0.50 of 11.8 tons. Supposing a gun cartridge
to be rammed home to the density of water, and entirely converted into gas
before the projectile began to move, the pressure in the bore would rise to
43 tons per square inch at the breech, and fall towards the muzzle, as the
travel of the shot afforded increasing room for expansion behind it. The
column of metal to be moved, even in the heaviest projectiles yet known, is
only a few pounds to the square inch of base, while the.maximum pressure
of the powder gas is measured in tons; it is clear therefore that the shot must
get under way at some period antecedent to the setting up of the maximum
pressure. In a breech-loader, where the projectile has to be forced through
a bore slightly less than its greatest diameter, it will be detained longer than
in a muzzle-loader, where it moves freely away, but the difference is insig-
nificant as regards the present argument. The result of the shot's early mo-
tion is that space is at once given for expansion, and the normal 43 tons is
never reached. Before these matters were fully understood, badly propor-
tioned charges of violent powder were found sometimes to set up what are
known to artillerists as "wave pressures," which were dynamical in charac-
ter, being caused by rushes of gas from one end of the charge to the other,
so that the gauges indicated far higher pressures at the ends of the powder
chamber than in the centre. This has now been overcome, and a great in-
crease of both power and safety has been obtained. Several important im-
provements have been made of late years; the principal ones are three in
number: — (1) a great stride was made in the manufacture of powder when
pebbles, prisms, and li inch cubes were introduced; (2) the discovery of the
beneficial effect of ''chambering," that is of boring out the powder chamber
to a greater diameter than the rest of the bore; (3) the method of ''air spac-
ing" the cartridge, so that a certain weight of powder should have a certain
definite space allotted to it, irrespective of the actual volume of the powder
grains. Thus in the 80 ton gun powder cubes of H inch edge are used, having
an absolute density a little over 1.75, or about 15.7 cubic inches to the pound.
ff these grains were rammed tightly home in a silk-siioiub bag. the space occu-
pied behind the shot would be 24.6 cubic inches per pound; as actually used,
an air-space over and within the cartridge is left, so that the space behind
the shot amounts to 34 cubic inches per pound. This density would set up a
pressure in a closed vessel of 26.6 tons per square inch, but the relief afforded
by the shot's motion reduces it to about 19 tons per square inch. The effect
of chambering out the end of the bore where the powder lies is practically
|:o permit a small gun to consume effectively the charge of a larger one. The
«*artridge is shortened, and the mechanical conditions of burning are greatly
sraproved, so that with large charges, higher velocity with lower pressure is
obtained from a chambered than from an unchambered gun. The above in-
formation is derived from the indications of the crusher gauge, which regi.^-
ters the pressure of the gas at various parts of the bore. The chronoscope
measures the rate at which the projectile acquires velocity during its travel
from the breech to the muzzle. Knowing the increment of velocity at any
point, we can calculate the amount of pressure required to produce this in-
crement, and thus confirmation is obtained of the accuracy of the records
obtained by the crusher gauge. The following table gives the increase ob-
tained at successive stages in the developm.ent of the power of the 80 too gun.
which was first under-bored for experiment, and gradually brought to it:^
present dimensions: —
i o
b
b
b
1— I
or
o 1
!
or
b
C?l
b
!
i"
1
j
00 pO C5 p 0»
b b b b b
? 1 ?
O 1 Ol
1
s
g- 5 i
li
5^
CC CO to »o to
c;t .-i. o to to
O O O j o o
1 1
3D
CO
*>-
^ pi p p ^
b b b b b
b
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It will be obserA^ed that each improvement has tended to facility of con-
Kumption of increased charges, so that while the pressures are diminished,
the penetrating power of the projectile is augmented, a heavier and more
destructive shell being driven through thicker armor.
The manner in which the various principles, of which an explanation has
been above attemjjted, are practically applied is laborious and complicated:
the conditions are often conflicting, and the ultimate dimensions of a piece
of ordnance is commonly a compromise. A couple of simple examples will
illustrate the modus operandi. Suppose that it is required to design a gun
which shall not exceed a given length, but shall throw a projectile capable
of piercing a given thickness of iron at a given range. There are several
formulas of a more or less empirical nature for calculating the perforating
power of a projectile moving with a known velocity. Penetration is by some
regarded as a punching action, by some as a wedging action; probably it is a
compound of the two. Recent experiments carried on with the very high
velocity of about 2000 f. s. ha ? thrown some doubt on the soundness of any
of the formulai, That generally used in England is as follows: —
Let W = weight of projectile in tons;
r =r radius of projectile in inches;
V = velocity of projectile in feet per second;
.- E= — = energy of projectile in foot tons;
- t = thickness of plate perforated in inches;
then -— z=2.52.Y 1:«,
2 r*
This formula tells the gun-designer what energj- is necessary to overcome
the resistance of the jjlate. Guided by experience he assumes for the moment
a striking velocity; the other component of the energy, the weight of the pro-
jectile, is then directly arrived at. The proportions of armor- piercing shell
are the same for all guns so that the weight guides the dimensions, and the
calibre of the gun follows. Should this appear to be in no way unsuitable to
the length already laid down in the conditions, the gun-designer calculates
the loss of velocity in the given range and from the striking velocity deduces
the muzzle velocity and the muzzle energy. The excellent labors [Eesearches
OH Explofiivef<) of Captain A. Noble. F. R. S., of Elswick. and Professor Abel.
F. R. S.. have shown how to calculate the amount of work done by a pound
of jDowder for every volume of expansion its gas undergoes; theresults of
many c ireful experiments and much intricate calculation are embodied in
the table on page 165, Text Book of Gunnery, Captain G. Mackinlay, R. A.,
which affords the means of determining the tot.ii work performed by any
charge in any gun. A certain portion of this work is expended in heating
the gun and projectile in giving rotation and so forth: the remainder appears
as the energy of translation of the shot on leaving the muzzle. Large guns
realize a greater proportion of the total work than small ones: the gunmaker
knows very approximately by experience what percentage may be expected
from certain classes of ordnance with certain descriptions of powder.
Roughly the factor of effect may be put within the following limits: — boat
guns, 60 to 65 per cent.: medium guns (such as the 60 pdr. and 80 pdr. B. L.
R's., and the 8 in. M. L. R.), 70 to 80 per cent.; heavy guns. 85 to 95 per cent.
The method of calculation will be best understood from an example. Sup-
pose a charge of 425 fos. of P.y powder is to be tired from the 80 ton gun cham-
bered to 18 inches diameter: the projectile weighs 1700 tt)s.. and the space
behind it is 14, 450 cubic inches. The whole content of the bore is 60, 400
cubic inches, and the volume of the charge is (425. X 27.7) 11, 773 cubic inches,
the number of expansions therefore is 5.13; the table (page 165, Mackinlay.)
shows that powder gas expanding to this extent from a density equal to that
of w^ater can perform work amounting to 92.4 foot tons per lb. Since, how-
ever, the charge burns up in and has to fill (425 X 84) 14, 450 cubic inches
before doing work, the energy due to this extent of expansion (1.227) is lost,
and 17.7 foot tons per ft. must be deducted, leaving (92.4 — 17.7) 74.7 foot
tons per ft. as the total work the charge is capable of performing under these
conditions. It is known from the preliminary tests of the powder that in the
so ton gun between 92 and 95 per cent, of the total work will be realized.
Hence the energy of the projectile will lie between 29, 210 and 30, 155 foot
tons, and its muzzle velocity betueen 1581 and 1600 f. s. A reference to the
table in this article will show that the result actually arrived at lies nearly
midway between these limits. In this manner the charge required to impart
the necessary energy to a shot of given weight in a given length of bore, and
conversely, the length of bore which will contain the requisite number of
expansions of a given charge, are easily found; hence the charge required to
produce the necessarj^ energy is readily found; the air-space and the dimen-
sions of the powder chamber follow, and the inside of tJie gun is settled. The
gun-designer now has to put walls round his bore.
Guided by the knowledge previously mentioned as derived from the crusher
gauge and the chroaoscope, he lays do wa the pressure-; at each point of the
interior, and calculates the amount and strength of the metal to be used,
according to the sj)ecial system of construction employed, and thus the ex-
terior of the gun is settled.
To give another instance: — let it be required to make a gun of moderate
power, not to exceed 17000 1t;s. in weight: recoil (of piece only) must not
exceed 15 feet per second; the requirements are further, that the gun must
have a range of 1300 yards with not more than 2'^- elevation, and 4000 yards
with not more than 8- elevation. A gun weighing 17000 tt^s. and recoiling
at 15 f. s. velocity gives 255.000 units of momentum (in pounds and foot-
seconds): the iuitial momentum of the gun ic lecoiling is practically equal
to the momentum of the shot on leaving the muzzle. Hence the shell must
have 255.000 units of momentum, which may be composed of high velocity
and light weight, or low velocity and heavy weight. For general service,
accuracy and penetration are essential qualities: a shell varying in length
from tvv-o and a half to three times its diameter v.'iii be suitable for ordinary
purposes. We have now the following problem in ballistics, viz.: To find
the calibre of a sljell of the proper length, of sucli a weight that, with the
muzzle velocity required to give a range of 1300 yards at 2- elevation, the
muzzle momentum shall be 255.000 units. This problem is readily solved by
the methods indicated in chapter on trajectories and it is found that a shell
8 in. in diameter, weighing 180 t^ s.. and halving a muzzle velocity of 1400 f s..
will about fill the requirements. Proceeding in the same manner we find
that a projectile of the same dimensions and having the same muzzle velocity
will satisfy the second requirement. The calibre being thus settled, the pro-
portions of the piece remain to be worked out. There is little difficulty in
obtaining a velocity of 1400 f. s.. with a moderate charge and moderate pres-
sures, so the length of the bore and the disposition of the metal can be adjust-
ed to suit the strain of discharge. In the gun under consideration, the bore
should be made as long as possible, and the weight of metal thrown as far
forward as possible consistently with preserving due strength at the breech.
The reasons for this are twofold: — first, the longer the bore the less is the
breech pressure required to produce a given, muzzle velocit3^ and the less is
the maximum strain thrown upon the rotating agent; secondly, the more for-
ward the general disposition of the metal, the farther from the breech end
will be the cetre of gravity, and consequently the trunnions — a position which
favors steady shooting and absence of jump.
[From, Gunmaking and Gunnery, Col. E. Maitland, R. A. Encyclopedia
BritannicaJ.
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